, God is at the end.” -Max Planck “An equation for me has no meaning, unless it represents **a** thought of God.” - **Srinivasa** **A**. **Ramanujan** “ There is **a** higher power, not influenced by our wishes, which finally decides and judges.” -Werner Heisenberg I have endeavored to/ THAT THE UNIVERSE WAS CREATED FROM NOTHING. ERNST BORIS CHAIN (1906 - 1979) ERNST BORIS CHAIN (1906 - 1979) AWARDED **A** 1945 NOBEL PRIZE IN MEDICINE FOR HIS WORK WITH PENICILLIN. “The principle of divine purpose… stares the biologist in the face /

No number has captured the attention and imaginations of people throughout the ages as much as the ratio of **a** circle’s circumference to its diameter. 3.1415926535897932384626433832795028841971693993751058209749445923078... Copyright Audrey Weeks 2003 More Attempts to rationalize (all/ 450 AD Ptolemy (Alexandria, Egypt) 150 AD Also used by Columbus on his voyage to the New World **Srinivasa** **Ramanujan** (India, 1887-1920) (http://www.science-frontiers.com/sf053/sf053p19.htm) (This is an irrational approximation.) /

on white card, print off each set of text on **a** light coloured card; one shade for set 1, another for set 2. The cards are aligned here; names, dates, pictures and text all match up as laid out here. Andrew Wiles Stephen Hawking Sir Isaac Newton Blaise Pascal Charles Babbage **Srinivasa** **Ramanujan** Alan Turing Roger Penrose Kurt Godel Most Famous For… Something/

**Srinivasa** Perumal Tirumanakolam (Wedding Posture) Dhasavathara m Statues at Dhasava thara Mandap am Arulmigu Moolaigarudan Sannadhi at northeastern side of the temple terrace Dhasavatharam Art in Herbel Painting Viswaroopa Dharsan Art in Herbel Painting at Yekadhasi Mandapam (Vaikunda Vaasal Mandapam) O M N **A** M O N **A** R **A** Y **A** N **A**/else, O lord of the meeting rivers. by Basavanna Translated by **A**. K. **Ramanujan** The Bhakti Movement 800 **A**.D - 1700 **A**.D. Bhakti movement is responsible for the many rites and rituals /

explanation: Why do the fundamental particles behave in ways that are comprehensible and predictable through laws and equations? “An equation for me has no meaning, unless it represents **a** thought of God.” - **Srinivasa** **A**. **Ramanujan** An alternative worldview: better "It is good to be constantly reminded of the fact that science as we know it today is not inescapable and that we may construct/

Committee, ICM 2010, University of Hyderabad August 19-27, 2010. International Conference on “The Legacy of **Srinivasa** **Ramanujan**” University of Delhi December 17- 22, 2012. 15 th Galway Topology Colloquium Mathematical Institute, University of Oxford, 2429 St. Giles’,/co-operation among the departmental teachers. We also have good relation with our present and Ex-students. We have **a** departmental library. **A** good nos. of books in our library. Students are obedient and very keen to learn. We have internet/

MacMahon (1854 – 1929) to write down **a** table of values of p(n) for n from 1 to 100. **A** Historical Sidenote And it was MacMahon’s table, broken up into groups of five values each, which led **Srinivasa** **Ramanujan** (1887 – 1920) to his discoveries about / with odd exponents! Euler’s Famous Discovery That means we have Euler realized that this way of writing D(q) had **a** never-before-seen partition-theoretic interpretation! Euler’s Famous Discovery Theorem: For all positive integers n, the number of partitions of/

Signed number Topology Graphs etc. RENE DESCARTES (FRENCH) B:31-03-1596 D:11-02-1650 **A** MATHEMATICIAN IS **A** BLIND MAN IN **A** DARK ROOM SEARCHING FOR **A** BLACK CAT WHICH IS NOT THERE ! EVARISTE GALOIS (French) B:25-10-1811 D:31-05/ OF ALL TIMES ! INDIAN MATHEMATICAL WIZARD Son of **a** petty accountant in **a** cloth shop. His works were mainly on His works were mainly on Theory of partitions Continued fractions Continued fractions Infinite integrals etc. **SRINIVASA** **RAMANUJAN** B:22-12-1887 D:26-04-1920 GREAT /

before Newton. Blasé Pascal – 1623-1662 Blasé Pascal – 1623-1662 He was **a** child genius. wrote **a** paper on conic sections at his 16 years. Invented the mechanical calculator at 19! Formulated **a** law – that says the pressure applied on the fluid in **a** container is distributed equally. Called Pascal’s law. **Srinivasa** **Ramanujan** (1887-1920) BBBBorn in December 22nd in Erode, Tamil/

**a** ≥ b ≥ c **a** ≥ c ≥ b b ≥ **a** ≥c b ≥ c ≥**a** c ≥ **a** ≥ b c ≥ b ≥ **a** Continued on next slide Proof by Cases Case 1: **a** ≥ b ≥ c (**a** @ b) = **a**, **a** @ c = **a**, b @ c = b Hence (**a** @ b) @ c = **a** = **a** @ (b @ c) Therefore the equality holds for the first case. **A** / odd is similar. The use phrase without loss of generality (WLOG) indicates this. Existence Proofs Proof of theorems of the form . **Srinivasa** **Ramanujan** (1887-1920) Proof of theorems of the form . Constructive existence proof: Find an explicit value of c, for which P(c)/

Bangalore S.E.T.S. Chennai … French Science Today 21 Madras Kolkata BangaloreChennai Kanpur Kumbakonam **Srinivasa** **Ramanujan** 22 http://www.isical.ac.in/ Statistics and Mathematics Unit, Kolkata Applied Statistic Division 23 http/ (Enigma) 56 Colossus Max Newman, the first programmable electronic computer (Bletchley Park before 1945) 57 Information theory Claude Shannon **A** mathematical theory of communication Bell System Technical Journal, 1948. 58 Claude E. Shannon " Communication Theory of Secrecy Systems ", /

Bangalore S.E.T.S. Chennai … French Science Today 21 Madras Kolkata BangaloreChennai Kanpur Kumbakonam **Srinivasa** **Ramanujan** 22 http://www.isical.ac.in/ Statistics and Mathematics Unit, Kolkata Applied Statistic Division 23 http/ (Enigma) 55 Colossus Max Newman, the first programmable electronic computer (Bletchley Park before 1945) 56 Information theory Claude Shannon **A** mathematical theory of communication Bell System Technical Journal, 1948. 57 Claude E. Shannon " Communication Theory of Secrecy Systems ", /

Saha (1893–1956), P. C. Mahalanobis (1893–1972), Sir C. V. Raman (1888– 1970), Subrahmanyan Chandrasekhar (1910–1995), Homi Bhabha (1909–1966), **Srinivasa** **Ramanujan** (1887–1920), Vikram Sarabhai (1919–1971), Hargobind Khorana (1922–), and Harish Chandra (1923–1983) are **a** few of the notable scholars of this period. [118] [118]Sir Jagadis Chandra BoseSatyendra Nath BoseMeghnad SahaP. C. MahalanobisSir C. V. RamanSubrahmanyan Chandrasekhar/

**a** century that saw two world wars with millions of dead in each), faster and faster computers were built and (more importantly, perhaps) better and better algorithms were developed. One of the best early ones is due to the great Indian mathematician **Srinivasa** **Ramanujan**/= 0. Sums, products, differences, quotients, and roots of algebraic numbers are algebraic. Transcendental Numbers By definition, **a** real or complex number is transcendental if (and only if) it is not algebraic. In 1851 the French mathematician /

Kanpur Indian Institute of Science Bangalore S.E.T.S. Chennai … French Science Today 21 Madras Kolkata BangaloreChennai Kanpur Kumbakonam **Srinivasa** **Ramanujan** 22 http://www.isical.ac.in/ Statistics and Mathematics Unit, Kolkata Applied Statistic Division 23 http://www.isical.ac.in/for cryptography Computing modulo n means working in the multiplicative group (Z/nZ) Specific attacks have been developed, hence **a** group of large size is required. We wish to replace this group by another one in which it is easy /

they are useful for improving the reliability of data storage media as well as to correct errors cause when **a** hard drive fails. The National Aeronautics and Space Administration (NASA) has used many different error-correcting codes for/: Finite fields theory Evariste Galois (1811-1832) Construction of regular polygons with rule and compass Group theory **Srinivasa** **Ramanujan** (1887-1920) 95 Coding Theory transmission SourceCoded Text Receiver 96 Error correcting codes Transmission with noise SourceCoded Text/

since 1729 = 10 3 + 9 3 = 12 3 + 1 3 Godfrey Harold Hardy (1877-1947) **Srinivasa** **Ramanujan** (1887-1920) Nonconstructive Existence Proofs In **a** nonconstructive existence proof, we assume no c exists which makes P(c) true and derive **a** contradiction. Example: Show that there exist irrational numbers x and y such that x y is rational. Proof: We know that √2 is irrational/

since 1729 = 10 3 + 9 3 = 12 3 + 1 3 Godfrey Harold Hardy (1877-1947) **Srinivasa** **Ramanujan** (1887-1920) Nonconstructive Existence Proofs In **a** nonconstructive existence proof, we assume no c exists which makes P(c) true and derive **a** contradiction. Example: Show that there exist irrational numbers x and y such that x y is rational. Proof: We know that √2 is irrational/

at least 3 Rate : 2/5. 2 data bits, 3 check bits Length 5 104 4 code words: **a**, b, **a**, b, **a**+b Each has 5 neighbours Each of the 4 balls of radius 1 has 6 elements There are 24 possible / algebraic equations with radicals: Finite fields theory Evariste Galois (1811-1832) Construction of regular polygons with rule and compass Group theory **Srinivasa** **Ramanujan** (1887-1920) 155 Codes and Mathematics Algebra (discrete mathematics finite fields, linear algebra,…) Geometry Probability and statistics 156 Codes and /

at least 3 Rate : 2/5. 2 data bits, 3 check bits Length 5 104 4 code words: **a**, b, **a**, b, **a**+b Each has 5 neighbours Each of the 4 balls of radius 1 has 6 elements There are 24 possible/ algebraic equations with radicals: Finite fields theory Evariste Galois (1811-1832) Construction of regular polygons with rule and compass Group theory **Srinivasa** **Ramanujan** (1887-1920) 144 Codes and Mathematics Algebra (discrete mathematics finite fields, linear algebra,…) Geometry Probability and statistics 145 Codes and /

: dreamed by Mendeleyev and August Kekulé Chemical transmision of nerve impulses: Otto Loewi Mathematics intuiton+dreams: **Srinivasa** **Ramanujan** The son of an Orthodox priest, Tesla was **a** mystic who had out-of-body experiences from the time he was seven years old. As **a** child, he saw **a** photograph of Niagara Falls and prophesied that one day he would harness the power there. Nikola Tesla/

with radicals: Finite fields theory Evariste Galois (1811-1832) Construction of regular polygons with rule and compass Group theory **Srinivasa** **Ramanujan** (1887-1920) 97 Coding Theory transmission SourceCoded Text Receiver 98 Error correcting codes Transmission with noise SourceCoded Text Receiver 99/ This is an error correcting code on the alphabet {0, 1, 2} with rate 1/2 Rule: **a** b **a**+b **a**+2b modulo 3 149 Sphere Packing While Shannon and Hamming were working on information transmission in the States, John /

algebraic equations with radicals: Finite fields theory Evariste Galois (1811-1832) Construction of regular polygons with rule and compass Group theory **Srinivasa** **Ramanujan** (1887-1920) 21 Codes and Mathematics Algebra (discrete mathematics finite fields, linear algebra,…) Geometry Probability and statistics 22 Codes and /distance at least 3 Rate : 2/5. 2 data bits, 3 check bits Length 5 121 4 codewords: **a**, b, **a**, b, **a**+b Each has 5 neighbours Each of the 4 balls of radius 1 has 6 elements There are 24 possible/

algebraic equations with radicals: Finite fields theory Evariste Galois (1811-1832) Construction of regular polygons with rule and compass Group theory **Srinivasa** **Ramanujan** (1887-1920) 12 Codes and Mathematics Algebra (discrete mathematics finite fields, linear algebra,…) Geometry Probability and statistics 13 Codes and /distance at least 3 Rate : 2/5. 2 data bits, 3 check bits Length 5 112 4 codewords: **a**, b, **a**, b, **a**+b Each has 5 neighbours Each of the 4 balls of radius 1 has 6 elements There are 24 possible/

integer that can be written as the sum of cubes of positive integers in two different ways: Proof: 1729 is such **a** number since 1729 = 10 3 + 9 3 = 12 3 + 1 3 Godfrey Harold Hardy (1877-1947) **Srinivasa** **Ramanujan** (1887-1920) 56 Nonconstructive Existence Proofs In **a** non-constructive existence proof, we assume no c exists which makes P(c) true and derive/

: dreamed by Mendeleyev and August Kekulé Chemical transmision of nerve impulses: Otto Loewi Mathematics intuiton+dreams: **Srinivasa** **Ramanujan** The son of an Orthodox priest, Tesla was **a** mystic who had out-of-body experiences from the time he was seven years old. As **a** child, he saw **a** photograph of Niagara Falls and prophesied that one day he would harness the power there. Nikola Tesla/

since 1729 = 10 3 + 9 3 = 12 3 + 1 3 Godfrey Harold Hardy (1877-1947) **Srinivasa** **Ramanujan** (1887-1920) 56 Nonconstructive Existence Proofs In **a** nonconstructive existence proof, we assume no c exists which makes P(c) true and derive **a** contradiction. Example: Show that there exist irrational numbers x and y such that x y is rational. Proof: We know that √2 is/

distance at least 3 Rate : 2/5. 2 data bits, 3 check bits Length 5 93 4 codewords: **a**, b, **a**, b, **a**+b Each has 5 neighbours Each of the 4 balls of radius 1 has 6 elements There are 24 possible/algebraic equations with radicals: Finite fields theory Evariste Galois (1811-1832) Construction of regular polygons with rule and compass Group theory **Srinivasa** **Ramanujan** (1887-1920) 144 Codes and Mathematics Algebra (discrete mathematics finite fields, linear algebra,…) Geometry Probability and statistics 145 Codes and /

: dreamed by Mendeleyev and August Kekulé Chemical transmision of nerve impulses: Otto Loewi Mathematics intuiton+dreams: **Srinivasa** **Ramanujan** The son of an Orthodox priest, Tesla was **a** mystic who had out-of-body experiences from the time he was seven years old. As **a** child, he saw **a** photograph of Niagara Falls and prophesied that one day he would harness the power there. Nikola Tesla/

since 1729 = 10 3 + 9 3 = 12 3 + 1 3 Godfrey Harold Hardy (1877-1947) **Srinivasa** **Ramanujan** (1887-1920) Nonconstructive Existence Proofs In **a** nonconstructive existence proof, we assume no c exists which makes P(c) true and derive **a** contradiction. Example: Show that there exist irrational numbers x and y such that x y is rational. Proof: We know that √2 is irrational/

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